All kinds of patterns

Mathematical equations, Turing felt, could explain phenomena as different as the stripes of zebra to patterns in sand
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First Published: Thu, Nov 22 2012. 05 58 PM IST
If you expand your notion of patterns, you will find reaction-diffusion systems that produce them in all kinds of intriguing places. Photo: Ali Mazraie Shadi/Thinkstock
If you expand your notion of patterns, you will find reaction-diffusion systems that produce them in all kinds of intriguing places. Photo: Ali Mazraie Shadi/Thinkstock
One low-tide evening some months ago, my school chum Ajay and I found ourselves strolling along a pristine beach near Alibag. We were chatting about kids and work and long-ago crushes—the usual when pals meet. Then we came to a spot where the receding waves had left marks in the sand that I’m sure are familiar to you: A series of dark, wavy lines each about a foot apart. The colour was probably from some oily residue in the water; indeed, our host that weekend, our school-chum Priya, had mentioned the oil problem.
Here, the colour made for a striking pattern, reminiscent of the coat of a tiger. They’re so pretty, I said to Ajay idly, but I’ve never understood how these are formed. I mean, I know the waves did it, but why the patterns?
His eyes lit up. “Aha!” he said. (He’s the only man I know who actually uses that word). “Turing patterns! And I’ve been using them in my own work!” Which surprised me, because Ajay is a biologist who leads a research lab. What was he doing looking at Turing patterns?
When we sat down over a cup of tea a little later, I gave him my shirt pocket pad and a pen. He sketched out for me a little world I had only the vaguest idea of, at the intersection of computer science and biology.
Alan Turing is one of the great figures of computer science. My last column here spoke of his ideas on making computers think; and even earlier, his Turing machine thought experiment laid the theoretical foundation for the science of computers—yes, I mean even that thing you use to send texts. This and plenty more about Turing, I remember from my days studying the subject. But what I didn’t remember about him is that in 1952, not long before he tragically committed suicide, he published a path-breaking paper called The Chemical Basis of Morphogenesis. In it, he laid out a hypothesis for the way certain patterns form in nature.
Imagine, for example, an “activator” chemical that stimulates the production of a skin pigment in an animal. Imagine an “inhibitor” chemical that hinders the production of the same pigment. (Such chemicals exist). Imagine now an embryo of the animal, through whose cells these chemicals diffuse, in tandem, reacting to each other as they move.
In that paper, Turing showed mathematically that such a “reaction-diffusion” system could, over time, produce some startling phenomena. For example, it could cause a regular variation in the concentration of the pigment deposits. This would explain the presence of stripes—as on zebras or cheetahs, for example. Reacting in different ways, it would explain the reticulated pattern you see on a giraffe. More generally, Turing’s mathematical simulation of reaction-diffusion systems suggested that this mechanism lay beneath the formation of many kinds of patterns in nature. In his paper, Turing speculated that it applied in two biological examples: “tentacle patterns on Hydra”—a tube-shaped jellyfish with tentacles around its mouth—“and whorled leaves”.
But the startling thing—and here’s more reason to love mathematics—is that if you expand your notion of patterns, you will find reaction-diffusion systems that produce them in all kinds of intriguing places.
Think of a large forest with a population of small antelopes, let’s say, and a population of tigers that prey on the antelopes. Left to themselves, the antelopes are like activators, steadily increasing their population. But the tigers are inhibitors, because they regularly eat antelopes. Over time, as these two populations react to each other, Turing’s ideas suggest that you will get areas where tigers predominate, interleaved with areas where antelopes predominate.
And Ajay quickly sketched for me yet another possible pattern. In the embryos of certain small fish, the delicate tango of inhibitor and activator chemicals ends up producing, of all things, rows of regularly spaced sensory hairs. The fish use these to gauge things about the water that surrounds them.
This is, of course, my necessarily limited understanding of the diagrams and arrows and words Ajay filled my pad with. But it also filled my mind with all kinds of leads I would love to explore.
Though I wonder about that. Turing called the mathematics in his paper “relatively elementary”. This is a serious blow to the ego for guys like me who think we understand things mathematical. The “elementary” equations he uses, the way he easily manipulates them, it all soars swiftly beyond my pretensions. They remind me forcefully that I’m just a dabbler.
Still, even I can understand the spirit of his paper. And so I’m trying to work out how those graceful patterns form on the beach. What in the water, or in the sand itself, acts as activator and inhibitor?
Exactly what is activated and inhibited? I have no idea. And yet these questions are why there’s joy in all this. So tell me your ideas about beach patterns.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences.
To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza
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First Published: Thu, Nov 22 2012. 05 58 PM IST
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